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%I #20 May 24 2020 04:24:32
%S 1,2,2,6,11,6,24,60,60,24,120,366,501,366,120,720,2532,4242,4242,2532,
%T 720,5040,19764,38268,46863,38268,19764,5040,40320,172512,373104,
%U 528336,528336,373104,172512,40320,362880,1668528,3942108,6237828,7213761,6237828,3942108,1668528,362880
%N Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial P(n) = n*(x + 1)*P(n - 1) - (n - 2)^2*x*P(n - 2).
%F P(0) = 0, P(1) = 1 and P(n) = n * (x + 1) * P(n - 1) - (n - 2)^2 * x * P(n - 2).
%e For n = 3, the polynomial is 6*x^2 + 11*x + 6.
%e The first few polynomials, as a table:
%e [1],
%e [2, 2],
%e [6, 11, 6],
%e [24, 60, 60, 24],
%e [120, 366, 501, 366, 120]
%p P:= proc(n) option remember; expand(`if`(n<2, n,
%p n*(x+1)*P(n-1)-(n-2)^2*x*P(n-2)))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(P(n)):
%p seq(T(n), n=1..12); # _Alois P. Heinz_, Jan 31 2018
%p A := proc(n,k) ## n >= 0 and k = 0 .. n
%p option remember;
%p if n = 0 and k = 0 then
%p 1
%p elif n > 0 and k >= 0 and k <= n then
%p (n+1)*(A(n-1,k)+A(n-1,k-1))-(n-1)^2*A(n-2,k-1)
%p else
%p 0
%p end if;
%p end proc: # _Yu-Sheng Chang_, Apr 14 2020
%t P[n_] := P[n] = Expand[If[n < 2, n, n (x+1) P[n-1] - (n-2)^2 x P[n-2]]];
%t row[n_] := CoefficientList[P[n], x];
%t row /@ Range[12] // Flatten (* _Jean-François Alcover_, Dec 10 2019 *)
%o (Sage)
%o @cached_function
%o def poly(n):
%o x = polygen(ZZ, 'x')
%o if n < 1:
%o return x.parent().zero()
%o elif n == 1:
%o return x.parent().one()
%o else:
%o return n * (x + 1) * poly(n - 1) - (n - 2)**2 * x * poly(n - 2)
%Y Very similar to A298854.
%Y Row sums are A277382(n-1) for n>0.
%Y Leftmost and rightmost columns are A000142.
%Y Alternating row sums are A177145.
%Y Alternating row sum of row 2*n+1 is A001818(n).
%K tabl,nonn,easy
%O 1,2
%A _F. Chapoton_, Jan 31 2018
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